158 Proceedings of the Royal Society of Edinburgh. [Sess. 
from which two equalities there results by multiplication 
Gq + CO 1 Cl . 2 + ... +0) Cl j + wd<2 + . . . + ^ 
b 4 -f- w&g + • • • + co 4 6g + oo 1 &2 4* ... +oo 
This being independent of the cc’s furnishes a factor of the eliminant, and 
the factor thus reached is seen to be the quadratic factor specified in the 
enunciation of the theorem. 
Note is taken that when n is odd there is the linear factor 
a \ a 2 -t * • • + d n + £>i + & 2 + • • . + b n , 
and that when n is even there is in addition to this factor 
a ± -a 2 + ... -a n + b ± -b 2 + ... - b n . 
Further, the multiplications indicated in the quadratic factor are actually 
performed, and « made to disappear through substitutions like 2 cos f 7r for 
oo + oT 1 . 
Muir, T. ( 1881). 
[On circulants of odd order. Quart. Journ. of Math., xviii, pp. 261-265.] 
The main theorem here established is that the cofactor of a 4 + a 2 + . . . 
+ a 2n +i vn C(a 4 , a 2 , . . . , a 2n +i) is expressible as a symmetric determinant 
of the n th order whose distinct elements are the Jn(n + 1) differences of the 
o o O o 
cyclic sums EaJ, Na^, Na^g, . . . , Ea^. 
Calling these cyclic sums A 1 , A 2 , A 3 , A 4 , in the case where n is 3, and 
multiplying C(ciq , a 2 , . . . , a 7 ) by itself in the form obtained by diminishing 
each row by the immediately preceding row, we find 
4 
■^2 
CO 
A 
4 
A 4 
A3 
4 
A 2 - 
Ai — A 2 
A 2 ~ A3 
a 3 - 
4 
A 4 - A 3 
4.-4 
A3 — A 2 
a 2 - a 4 
A 4 - A 2 
a 9 - 
A 3 
00 
1 
> 
a 4 - A 3 
A 4 - A 3 
A3 ~ ^2 
A 2 - A 1 
aJ 
a 2 
A 9 — A 3 
A3 - A 4 
A 4 - A 3 
A3 — A 2 
a 2 - 
a| 
A x - A 2 
A 2 - A 3 
A3 ~ A 4 
a 3 -a 4 
CO 
1 
< 
A3 - 
a 2 
A 2 — Aj 
A 4 - A 2 
A 2 Ag 
A 2 “ A3 
1 
CO 
<1 
A 4 - 
a 3 
A3 ~ A 2 
A 2 - A x 
A 4 ~ A 2 
Here the increasing of the first column by all the others shows that 
A 1 + 2A 2 + 2A 3 + 2 A 4 , i.e. (cq + $ 2 + • . • + oqP 
is a factor, and the full result takes the form 
(C 7 ) 2 = («! + a 2 + ... 4 aff 
a 
p 
y 
— a 
a 
P 
y 
~P 
— a 
a 
P 
-y 
-P 
— a 
a 
-y 
-P 
- a 
y 
-y 
-P 
-r -P 
y 
P 
a 
— a 
-y 
y 
P 
a 
